Articles

In this paper we show the validity of Stein’s interpolation theorem on variable exponent Morrey spaces.

We establish a sharp Olsen type inequality $ \big \| g {\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}} } \leq C \big \| g \big \|_{L^{q}_{\ell } } \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $ for multilinear fractional integrals $$\mathcal{I}_{\alpha}(\overrightarrow{f})(x)=\int\limits_{(\mathbb{R}^{n})^{m}}\frac{f_1(y_1)⋯f_m(y_m)}{(|x−y_1|+⋯+|x−y_m|)^{mn-\alpha}}d}(\overrightarrow{y}, x∈R^n,$$ $0 < \alpha < mn$, where $L_r^q, L_l^q, L_{s_j}^{p_j}, j = 1,…,m,$ are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp because it gives necessary and sufficient condition on a weight function $V$ for which the inequality $ \big \|{\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}}(V) } \leq C \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $ holds. Morrey spaces play an important role in relation to regularity problems of solutions of partial differential equations. They describe the integrability more precisely than Lebesgue spaces. We also derive a characterization of the trace inequality $\big \| B_{\alpha } (f_{1},f_{2})\big \|_{{L^{q}_{r}}(d\mu ) } \leq C \prod\limits_{j=1}^{2} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}} ({\Bbb {R}}^{n}) }, $ in terms of a Borel measure μ, where Bα is the bilinear fractional integral operator given by the formula $Bα(f_1,f_2)(x)=\int\limits_{R^n}\frac{f_1(x+t)f_2(x−t)}{|t|^{n−α}}dt,0<α

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator $T_γ,μ$ defined with respect to a measure μ on a σ-algebra of Borel sets of quasi-metric space X from the product $L_{p}^{1}(X,μ)×⋯×L_{p}^{m}(X,μ)$ to $Lq(X,μ)$ is established. The related weak type inequality is also obtained. The derived results are used to get appropriate boundedness of $T_γ,μ$ in Morrey spaces defined with respect to a measure $μ$.